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Dynamical Slowdown of Polymers in Laminar and
Random Flows
Antonio Celani, Alberto Puliafito, Dario Vincenzi
To cite this version:
Antonio Celani, Alberto Puliafito, Dario Vincenzi. Dynamical Slowdown of Polymers in Laminar
and Random Flows.
Physical Review Letters, American Physical Society, 2006, 97, pp.118301.
�10.1103/PhysRevLett.97.118301�. �hal-00097741�
ccsd-00097741, version 1 - 22 Sep 2006
Dynamical slowdown of polymers in laminar and random flows
A. Celani,1 A. Puliafito,1 and D. Vincenzi2, 3
1
CNRS–INLN, 1361 Route des Lucioles, 06560 Valbonne, France
2
Max-Planck-Institut f¨ur Dynamik und Selbstorganisation, Bunsenstraße 10, 37073 G¨ottingen, Germany
3
Sibley School of Mechanical & Aerospace Engineering and LASSP, Cornell University, Ithaca, NY 14853 The influence of an external flow on the relaxation dynamics of a single polymer is investigated theoretically and numerically. We show that a pronounced dynamical slowdown occurs in the vicinity of the coil–stretch transition, especially when the dependence on polymer conformation of the drag is accounted for. For the elongational flow, relaxation times are exceedingly larger than the Zimm relaxation time, resulting in the observation of conformation hysteresis. For random smooth flows hysteresis is not present. Yet, relaxation dynamics is significantly slowed down because of the large variety of accessible polymer configurations. The implications of these results for the modeling of dilute polymer solutions in turbulent flows are addressed.
Phys. Rev. Lett. 97, 118301 (2006)
http://link.aps.org/abstract/PRL/v97/e118301
The dynamics of an isolated polymer in a flow field forms the basis of constitutive models for dilute polymer solutions [1, 2, 3]. The modeling of drag-reducing flows, for instance, requires an appropriate description of sin-gle polymer deformation in turbulent velocity fields [4]. In the last decade, major advances in fluorescence mi-croscopy offered the possibility of tracking isolated poly-mers both in laminar [3] and random flows [5]. The dy-namics of a polymer in thermal equilibrium with the sur-rounding solvent is commonly described in terms of nor-mal modes and relaxation times associated with them. The analytical form of the relaxation spectrum was first obtained by Rouse under the assumption that the poly-mer could be described as a series of beads connected by Hookean springs [6]. The Rouse model was subsequently improved by Zimm to include hydrodynamical interac-tions between segments of the polymer [7]. In Zimm’s formulation, the equations of motion are decoupled into a normal mode structure by preaveraging the distances between the beads over the distribution of polymer con-figurations. The relaxation time associated with the fun-damental mode, τ , determines the typical time that it takes for a deformed polymer to recover the equilibrium configuration in a solvent. The normal mode theory was confirmed by the analysis of the oscillatory motion of a DNA molecule immersed in a solvent and held in a par-tially extended state by means of optical tweezers [8]. An alternative approach to examine polymer relaxation con-sists in stretching a tethered DNA molecule in a flow and measuring its relaxation after cessation of the flow [9, 10]. The theoretical predictions for these experimental con-ditions are provided by the scaling theory [11] and the static dynamics formalism [12].
The aforementioned studies all consider the internal dynamics of a polymer floating in a solvent under the influence of thermal noise only — the interaction of the polymer with an external flow is not taken into account. One of the aspects highlighted by experiments is that
polymer dynamics in a moving fluid is strongly influenced by the carrier velocity field. The coil–stretch transition is the most noticeable example: as the strain rate exceeds a threshold value, the polymer undergoes a transition from the coiled, equilibrium configuration to an almost fully extended one [13]. Therefore, when a polymer is freely transported by a non-uniform flow, we expect that the time scales describing its dynamics may be significantly different from the Zimm time τ . Simple models for poly-mer stretching indeed suggest deviations from Zimm’s theory near the coil–stretch transition [14, 15, 16]. Dis-crepancies in the definition of the correct relaxation time are also encountered in drop formation experiments [17]. In this Letter we investigate polymer relaxation dy-namics both in elongational and random smooth flows. Our analysis brings to evidence an important slowdown of dynamics with respect to the Zimm timescales, in the vicinity of the coil–stretch transition. For the elon-gational flow, this is related to conformation hystere-sis [18, 19, 20]. For random flows, we show that hys-teresis is not present. Nonetheless, the amplification of the relaxation time persists, albeit to a lesser extent, due to the large heterogeneity of polymer configurations. In both cases, the dependence of the drag force on the poly-mer configuration plays a prominent role. This suggests the necessity of improving current models of polymer so-lutions in turbulent flows to account for such effect.
The dumbbell approximation is the basis of the most common models of single polymer dynamics and vis-coelastic models of dilute polymer solutions [21]. Its va-lidity relies on the fact that the slowest deformational mode of the polymer is the most influential in produc-ing viscoelasticity [4]. However, when attention is di-rected to non-equilibrium dynamics it is often too crude to assume that τ is independent of the conformation of the molecule [22]. Therefore, following de Gennes [13], Hinch [23], and Tanner’s [24] approach, we consider a model where the polymer is described as two beads con-Typeset by REVTEX
2 nected by an elastic spring and the probability density
function of the end-to-end vector, Ψ(R, t), satisfies the diffusion equation: ∂Ψ ∂t = − ∂ ∂R· κ(t)·R Ψ−f (R)R 2τ ν(R)Ψ− R2 0 2τ ν(R) ∂Ψ ∂R (1) where κij(t) = ∂jvi(t) is the velocity gradient, R0 is the
mean extension at thermal equilibrium and R = |R|. The function f (R) defines the entropic force restoring stretched molecules into the coiled configuration. Syn-thetic polymers are properly described by the Warner law, f (R) = 1/(1 − R2/L2), where L is the contour
length of the polymer; biological macromolecules are better characterized by the Marko–Siggia law, f (R) = 2/3 − L/(6R) + L/[6R(1 − R/L)2] [2]. The flow strength
relative to the polymer tendency to recoil is expressed by the Weissenberg number Wi, defined as the product of the Zimm time τ by a characteristic extension rate of the flow. The function ν(R) encodes for the depen-dence on the polymer conformation of the drag exerted by the fluid: a spherical coil offers a smaller resistance with respect to a long rod-like configuration. We utilize the expression ν(R) = 1 + (ζs/ζc− 1)R/L that
interpo-lates linearly between these extremes (see Refs. [25, 26]). Here, ζc = 3
√ 6π3R
0η/8 and ζs= 2πLη/ ln(L/ℓ) are the
friction coefficients for the coiled and the stretched con-figuration, respectively, η is the solvent viscosity and ℓ is the diameter of the molecule. Strictly speaking, the above form of ν(R) was deduced from experiments and microscopic simulations of laminar flows. To our knowl-edge, measures of ν(R) in random flows are not available yet, and the study of its functional dependence lies be-yond the scope of the present work. Assuming that the polymer is aligned with the local stretching direction of the flow for the most part of its evolution, we shall use a linear ν(R) for a random flow as well.
To define the relaxation time in the presence of an ar-bitrary external flow, we consider the probability density function of the rescaled extension, P (r, t) with r = R/L, and identify the dynamical relaxation time, trel, as the
characteristic time needed for P (r, t) to attain its station-ary form Pst(r). In the cases examined in this Letter, as
we shall see, the probability density function of r satisfies the Fokker–Planck equation
∂t′P = −∂r(D1(r)P ) + ∂rD2(r)∂rP, (2)
where the form of D1(r) and D2(r) depends on the
flow and t′
= t/τ . The stationary solution to Eq. (2) takes the potential form Pst(r) = N exp [−E(r)/KBT ],
where N is the normalization constant and E(r) = −KBTRrD1(ρ)/D2(ρ) dρ. The finite-time solution
ad-mits the expansion P (r, t′ ) = Pst(r) + ∞ X n=1 anpn(r) e−t ′/σ n, (3) 0 5 10 15 20 25 30 35 0 0.5 1 1.5 trel / τ Wi (a) ζs/ζc=2 ζs/ζc=1 0 50 100 150 200 250 300 350 400 450 1 2 3 4 5 6 7 8 tmax / τ ζs /ζc (b)
Figure 1: Elongational flow: (a) rescaled relaxation time vs. Wi for b = 400. The entropic force is given by the Warner law. For small Wi, trel/τ grows as 1/(1 − 2Wi), as can be seen
by replacing ˆf (r) with 1 in Eq. (2). For Wi ≫ Wicrit, the time
needed to reach the asymptotic regime is set by the time scale of the flow, γ−1, and t
rel/τ decreases as Wi−1. (b) Rescaled
maximum relaxation time tmax/τ vs. ζs/ζc(b = 400).
where the coefficients an are fixed by P (r, 0), pn(r) are
the eigenfunctions of the Fokker–Planck operator, and σn
are the reciprocals of its (strictly positive) eigenvalues, arranged in descending order (σn > σn+1). The
relax-ation time is thus defined as trel≡ σ1τ .
As a first example, we examine the steady planar elon-gational flow v = γ(x, −y). By assuming that the poly-mer extension in the x-direction is much greater than in the y-direction, it is easy to derive from Eq. (1) an equa-tion of the form (2) with D1(r) = Wi r − ˆf (r)r/[2ˆν(r)],
D2(r) = [2bˆν(r)]−1, ˆf (r) = f (rL), ˆν(r) = ν(rL),
b = L2/R2
0, and Wi = γτ [27]. For ζs = ζc, trel can be
computed from (3) by solving a central two-point con-nection problem for a generalized spheroidal wave equa-tion [16]. In the general case, ζs > ζc, we resorted to a
numerical computation based on the variation–iteration method of quantum mechanics [28]. In the vicinity of the coil–stretch transition (Wi = 1/2) trelshows a sharp
peak (Fig. 1). In this range of Wi there is a critical competition between the entropic force and the velocity gradient that makes the convergence time to the steady state extremely long. This effect is strongly enhanced by the conformation-dependent drag; the peak tmax
in-deed increases with ζs/ζc (Fig. 1). Those extremely long
relaxation times are intimately connected with the obser-vation of finite-time conformation hysteresis [18, 19, 20]. For large enough ζs/ζc, there is a narrow range of Wi
around the critical value for the coil–stretch transition where E(r) has a double well structure [18, 19, 20]. The barrier height separating the coiled and the stretched state is much greater than the thermal energy KBT , and
therefore the polymer remains trapped in its initial con-figuration for an exceptionally long time (Fig. 2).
The discovery of elastic turbulence has recently al-lowed the examination of single polymer dynamics in a
0 10 20 30 40 50 60 0 0.5 1 6 5 4 3 2 1 0 E / KB T r (a) -1 0 1 2 0 0.5 1 E / KB T r (c) -2 0 2 4 E / KB T (b)
Figure 2: (a) Effective energy at the coil–stretch transition for a polyacrylamide (PAM) molecule (b = 3953, ζs/ζc = 6.87)
in the elongational flow (dashed line) and the 3D Batchelor– Kraichnan flow (solid line). The left vertical axis refers to the dashed line; the right vertical axis refers to the solid line; (b) effective energy in the random flow for a PAM molecule with constant drag (ζs=ζc) (from top to bottom Wi =
0.4, 0.7, 1.0, 1.3); (c) the same as (b), but with ζs = 6.87ζc
(from top to bottom Wi = 0.28, 0.33, 0.38, 0.43). When ζs>
ζcthe transition occurs in a much narrower range of Wi.
random smooth flow generated by viscoelastic instabili-ties [5, 29]. The velocity gradient fluctuates along fluid trajectories; the average local deformation rates define the three Lyapunov exponents of the flow. Experimental observations have been accompanied by theoretical and numerical studies based on the dumbbell model (see, e.g., Refs. [14, 15, 30]). To analytically investigate polymer relaxation dynamics in random flows, we ini-tially assume that the velocity field obeys the Batchelor– Kraichnan statistics [31]. The velocity gradient is then a statistically isotropic and parity invariant Gaussian pro-cess with zero mean and correlation: hκij(t)κkl(s)i =
2λδ(t−s)[(d+1)δikδjl−δijδkl−δilδjk]/[d(d−1)], where d is
the dimension of the flow and λ denotes the largest Lya-punov exponent. In this context, we indicate by P (r, t′
) the probability density function of the extension both with respect to thermal noise and the realizations of the velocity field. For the elongational flow Eq. (2) was ob-tained under the uniaxial approximation. On the con-trary, for the isotropic random flow Eq. (2) holds ex-actly and can be obtained by a Gaussian integration by parts followed by integration over angular variables. The drift and diffusion coefficients take the form D1(r) =
(d − 1)/d Wi r − ˆf (r)r/[2ˆν(r)] + (d − 1)/[2bˆν(r)r] and D2(r) = Wi r2/d + [2bˆν(r)]
−1 with Wi = λτ . The
sta-tionary distribution admits once more a potential form. For d = 3 and ζs> ζc, the potential displays a very wide
well, the effect of the conformation-dependent drag being to increase the probability of large extensions. There is no evidence of pronounced double wells (Fig. 2). Near the coil–stretch transition, the effective barrier heights separating the coiled and stretched states are indeed at
most comparable to thermal energy. For realistic ζs/ζc
no conformation hysteresis is therefore expected to be observed in random flows. The behavior of trel vs. Wi
is however analogous to the one encountered in the elon-gational flow: trel/τ increases as [1 − Wi(d + 2)/d]−1 at
small Wi and decreases as Wi−1 at large Wi. A peak near the transition is present, that becomes more and more pronounced with increasing ζs/ζc, attaining values
as large as about thirty times τ (Fig. 3). The reason for this behavior is the breadth of Pst(r) and the consequent
large heterogeneity of accessible polymer configurations. To corroborate the results obtained in the context of the short-correlated flow, we performed Brownian Dy-namics simulations of dumbbell molecules [32] in the ran-dom flow introduced by Brunk et al. [34]. This model re-produces the small-scale structure of a turbulent flow by means of a statistically isotropic Gaussian velocity gra-dient. The autocorrelation times of components of the strain and rotation tensors are set to be multiple of the Kolmogorov time τη by comparison with direct
numer-ical simulations of 3D isotropic turbulence (τS = 2.3τη,
τR = 7.2τη). The Lyapunov exponent of this flow is
λ ≃ 10τ−1
η . We computed trel as the time of convergence
of the moments hrn(t)i to their stationary value hrni st:
t−1
rel = − limt→∞ln [hr
n(t)i − hrni
st]/t, where the
aver-ages were taken over an ensemble of realizations of the flow, all with the same initial extension r(0). The nu-merical difficulty arising from the singularity of the en-tropic force at R = L has been overcome by exploiting the algorithm introduced in [35]. The results shown in Fig. 4 confirm the scenario depicted in the context of the
0 5 10 15 20 25 30 0 0.5 1 1.5 trel / τ Wi (a) ζs/ζc=6.87 ζs/ζc=1 5 10 15 20 25 30 1 2 3 4 5 6 7 8 tmax / τ ζs/ζc (b)
Figure 3: 3D Batchelor–Kraichnan flow: (a) trel/τ vs. Wi
for a PAM molecule (b = 3953); (b) tmax/τ for the following
polymers: DNA (•, b = 191.5; ◦, b = 260; , b = 565; +, b = 2250), polystyrene (×, b = 673), polyethyleneoxide (PEO) (N, b = 1666), Escherichia Coli DNA (△, b = 9250), PAM (). Measures of b and ζs/ζccan be found in [18, 19, 20, 25].
Synthetic polymers are modeled by the Warner law, whereas biological molecules are described by the Marko–Siggia law. Relaxation times were computed by means of the variation– iteration method [28]. For ζs = ζc they can be obtained by
4 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 trel / τ Wi (a) ζs/ζc=4.47 ζs/ζc=1 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 trel / τ Wi (b) ζs/ζc=6.87 ζs/ζc=1
Figure 4: Brunk–Koch–Lion flow: trel/τ vs. Wi = λτ from
Brownian Dynamics simulations; (a) PEO; (b) PAM.
short-correlated flow. It is worth noting that the above definition provides an operational method to measure trel
that can be implemented in experiments.
In summary, we have shown that the equilibrium con-figuration of a polymer in a flow, as well as the time a de-formed molecule takes on average to recover that config-uration, depend sensitively on the properties of the flow. In the vicinity of the coil–stretch transition the character-istic relaxation time is much longer than the Zimm time, both in elongational and random flows. In other words, the effective Wi differs considerably from the “bare” one [36]. This effect is strongly amplified when the drag coefficient depends on the conformation of the polymer, and may play an important role in drag-reducing turbu-lent flows, where the strain rate often fluctuates around values typical of the coil–stretch transition [38]. Our con-clusions thus suggest that the conformation-dependent drag should be included as a basic ingredient of contin-uum models of polymer solutions, calling for further the-oretical, experimental and numerical study.
The authors gratefully acknowledge inspiring discus-sions with M. Chertkov, S. Gerashchenko, M. Martins Afonso and V. Steinberg. This work has been partially supported by the EU (contract HPRN-CT-2002-00300).
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